Is Bohmian Mechanics deterministic?

[atyy]

I've also heard that Bohmian mechanics is deterministic (eg. http://arxiv.org/abs/1206.1084, quote from Bell, p17).

But in all presentations (eg. http://arxiv.org/abs/1206.1084, p30) I've come across so far, Bohmian mechanics needs an initial probability distribution. If probability over the initial conditions is introduced, in what sense is Bohmian mechanics deterministic?

 

[Ilja]

The equations of motion of BM are deterministic. But you can, of course, define statistical distributions of initial values in deterministic theories, and obtain equations for the evolution of these statistical distributions too.

To obtain from BM the predictions of quantum theory, this is what you have to do - if you introduce the Born probability distribution as the initial value, you obtain, from the fundamental deterministic law of evolution, an
evolution equation for these initial values, and it follows that, in this case, the Born rule always holds.

There is also another point: Deterministic evolution may be chaotic. So, to make predictions, it may be necessary to apply statistical methods even if the fundamental equations are deterministic. If you do this, you can prove results like Valentinis subquantum H-theorem. This allows to prove that the Born rule is something
like a statistical equilibrium for quite arbitrary initial values.

You can also do actual computations for some test examples, and you find out that already for not very complicated systems this Born rule equilibrium will be reached very fast.

All this are statistical considerations which can be applied without questioning the trivial fact that the fundamental equations of BM are deterministic.

 

[atyy]

There is also another point: Deterministic evolution may be chaotic. So, to make predictions, it may be necessary to apply statistical methods even if the fundamental equations are deterministic. If you do this, you can prove results like Valentinis subquantum H-theorem. This allows to prove that the Born rule is something
like a statistical equilibrium for quite arbitrary initial values.

Thanks for the pointer to Valentini's work. Is it an attempt to make Bohmian mechanics deterministic even in the initial conditions - like the attempt to derive classical thermodyanamics for Anosov-like systems through the SRB measure http://arxiv.org/abs/0807.1268 (section V)?

 

[Ilja]

Thanks for the pointer to Valentini's work. Is it an attempt to make Bohmian mechanics deterministic even in the initial conditions - like the attempt to derive classical thermodyanamics for Anosov-like systems through the SRB measure http://arxiv.org/abs/0807.1268 (section V)?

I think the point was not determinism, but the aim to get rid of the necessity to postulate |ψ|² as an initial probability distribution, but to have it derived.

 

[Demystifier]

Atyy, Bohmian mechanics is (not) deterministic in the same sense in which classical statistical mechanics is (not) deterministic. In particular, initial conditions have some definite values, but there is no law which determines them. If someone does not know the initial conditions, then the best one can do is to predict a probability for a given initial condition.